Sakurai page 181: Time evolution of ensembles

Equation (3.4.27), at some time [itex]t_0[/itex], the density operator is given by[tex]
\rho(t_0) = \sum_i w_i \mid \alpha^{(i)} \rangle \langle \alpha^{(i)} \mid
[/tex]Equation (3.4.28), at a later time, the state ket changes from [itex]\mid \alpha^{(i)} \rangle[/itex] to [itex]\mid \alpha^{(i)}, t_0 ; t \rangle[/itex].

(Here ##|\alpha \rangle## is any ket and ##\langle \beta |## is any bra). If you apply this to your density operator you should get the desired result. Note that to evaluate the time derivative of a bra you will have to use the Hermitian conjugate of the Schrodinger equation.

If you want to make it totally transparent why the normal derivative product rule applies in this case, you can consider the infinitesmial time evolution of a ket:

If you write down the analagous formula for a bra, and then use the two formula to evaluate ##| \alpha(t+dt) \rangle \langle \beta(t+dt) |## then you should be able to derive the claimed product rule for derivatives (at the physicist level of rigor, anyway).